Applied Analysis Seminar 2 - Babuska Problem in Composite Materials and Some Applications
Date: Friday, October 22, 2021, 14:00-15:00
Speaker: Haigang Li (Beijing Normal University)
Zoom ID：361 038 6975 Password：BIMSA
Title: Babuska Problem in Composite Materials and Some Applications
In high-contrast fiber-reinforced composite materials, the stress concentration between two adjacent inclusions is a common phenomenon, which always causes damage initiation. For the original problem proposed by Ivo Babuska concerning the system of linear elasticity, we develop an iteration technique with respect to the energy integral to overcome the difficulty from the lack of maximal principle in PDE theory and obtain the blow-up asymptotic expressions of the gradients of solutions to the Lame system with partially infinite coefficients in the narrow region between inclusions when they are close to touch. Our results hold for convex inclusions with arbitrary shape and in all dimensions. As an application, we recently proved an extended Flaherty-Keller formula on the effective elastic property of a periodic composite with densely packed fibers, which is related to the “Vigdergauz microstructure” in the shape optimization of fibers. On the other hand, we recently applied our results to deal with the resonant behavior between two close-to-touching convex acoustic subwavelength resonators.